On the Complexity of Sets of Free Lines and Line Segments Among Balls in Three Dimensions
Identifieur interne : 002D33 ( Istex/Corpus ); précédent : 002D32; suivant : 002D34On the Complexity of Sets of Free Lines and Line Segments Among Balls in Three Dimensions
Auteurs : Marc Glisse ; Sylvain LazardSource :
- Discrete & Computational Geometry [ 0179-5376 ] ; 2012-06-01.
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- KwdEn :
Abstract
Abstract: We present two new fundamental lower bounds on the worst-case combinatorial complexity of sets of free lines and sets of maximal free line segments in the presence of balls in three dimensions. We first prove that the set of maximal non-occluded line segments among n disjoint unit balls has complexity Ω(n 4), which matches the trivial O(n 4) upper bound. This improves the trivial Ω(n 2) bound and also the Ω(n 3) lower bound for the restricted setting of arbitrary-size balls (Devillers and Ramos, personal communication, 2001). This result settles, negatively, the natural conjecture that this set of line segments, or, equivalently, the visibility complex, has smaller worst-case complexity for disjoint fat objects than for skinny triangles. We also prove an Ω(n 3) lower bound on the complexity of the set of non-occluded lines among n balls of arbitrary radii, improving on the trivial Ω(n 2) bound. This new bound almost matches the recent O(n 3+ε ) upper bound (Rubin, 26th Annual ACM Symposium on Computational Geometry—SCG’10, pp. 58–67, 2010).
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DOI: 10.1007/s00454-012-9414-8
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<front><div type="abstract" xml:lang="en">Abstract: We present two new fundamental lower bounds on the worst-case combinatorial complexity of sets of free lines and sets of maximal free line segments in the presence of balls in three dimensions. We first prove that the set of maximal non-occluded line segments among n disjoint unit balls has complexity Ω(n 4), which matches the trivial O(n 4) upper bound. This improves the trivial Ω(n 2) bound and also the Ω(n 3) lower bound for the restricted setting of arbitrary-size balls (Devillers and Ramos, personal communication, 2001). This result settles, negatively, the natural conjecture that this set of line segments, or, equivalently, the visibility complex, has smaller worst-case complexity for disjoint fat objects than for skinny triangles. We also prove an Ω(n 3) lower bound on the complexity of the set of non-occluded lines among n balls of arbitrary radii, improving on the trivial Ω(n 2) bound. This new bound almost matches the recent O(n 3+ε ) upper bound (Rubin, 26th Annual ACM Symposium on Computational Geometry—SCG’10, pp. 58–67, 2010).</div>
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<abstract lang="en">Abstract: We present two new fundamental lower bounds on the worst-case combinatorial complexity of sets of free lines and sets of maximal free line segments in the presence of balls in three dimensions. We first prove that the set of maximal non-occluded line segments among n disjoint unit balls has complexity Ω(n 4), which matches the trivial O(n 4) upper bound. This improves the trivial Ω(n 2) bound and also the Ω(n 3) lower bound for the restricted setting of arbitrary-size balls (Devillers and Ramos, personal communication, 2001). This result settles, negatively, the natural conjecture that this set of line segments, or, equivalently, the visibility complex, has smaller worst-case complexity for disjoint fat objects than for skinny triangles. We also prove an Ω(n 3) lower bound on the complexity of the set of non-occluded lines among n balls of arbitrary radii, improving on the trivial Ω(n 2) bound. This new bound almost matches the recent O(n 3+ε ) upper bound (Rubin, 26th Annual ACM Symposium on Computational Geometry—SCG’10, pp. 58–67, 2010).</abstract>
<subject lang="en"><genre>Keywords</genre>
<topic>3D visibility</topic>
<topic>Visibility complex</topic>
<topic>Free lines</topic>
<topic>Free segments</topic>
<topic>Balls</topic>
</subject>
<relatedItem type="host"><titleInfo><title>Discrete & Computational Geometry</title>
</titleInfo>
<titleInfo type="abbreviated"><title>Discrete Comput Geom</title>
</titleInfo>
<name type="personal"><namePart type="given">David</namePart>
<namePart type="family">Kirkpatrick</namePart>
<role><roleTerm type="text">editor</roleTerm>
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<genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre>
<originInfo><publisher>Springer</publisher>
<dateIssued encoding="w3cdtf">2012-03-27</dateIssued>
<copyrightDate encoding="w3cdtf">2012</copyrightDate>
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<subject><genre>Mathematics</genre>
<topic>Computational Mathematics and Numerical Analysis</topic>
<topic>Combinatorics</topic>
</subject>
<identifier type="ISSN">0179-5376</identifier>
<identifier type="eISSN">1432-0444</identifier>
<identifier type="JournalID">454</identifier>
<identifier type="IssueArticleCount">6</identifier>
<identifier type="VolumeIssueCount">4</identifier>
<part><date>2012</date>
<detail type="issue"><title>Special Issue: 26th Annual Symposium on Computational Geometry; Guest Editor: David Kirkpatrick</title>
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<detail type="volume"><number>47</number>
<caption>vol.</caption>
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<detail type="issue"><number>4</number>
<caption>no.</caption>
</detail>
<extent unit="pages"><start>756</start>
<end>772</end>
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<identifier type="DOI">10.1007/s00454-012-9414-8</identifier>
<identifier type="ArticleID">9414</identifier>
<identifier type="ArticleID">s00454-012-9414-8</identifier>
<accessCondition type="use and reproduction" contentType="copyright">Springer Science+Business Media, LLC, 2012</accessCondition>
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